Nonlocal Schrödinger-Poisson systems in ${\mathbb{R}}^N$: the fractional Sobolev limiting case

We study the existence of positive solutions for nonlocal systems in gradient form and set in the whole ${\mathbb{R}}^N$. A quasilinear fractional Schrödinger equation, where the leading operator is the $\frac{N}{s}$-fractional Laplacian, is coupled with a higher-order and possibly fractional Poisson equation. For both operators the dimension $N\ge 2$ corresponds to the limiting case of the Sobolev embedding, hence we consider nonlinearities with exponential growth. Since standard variational tools cannot be applied due to the sign changing logarithmic Riesz kernel of the Poisson equation, we employ a variational approximating procedure for an auxiliary Choquard equation, where the Riesz kernel is uniformly approximated by polynomial kernels. Qualitative properties of solutions such as symmetry, regularity and decay are also established. Our results extend and complete the analysis carried out in the planar case in [D. Cassani, Z. Liu, G. Romani. arxiv:2305.15274].